Express 
as the sum of two radicals of integers. (Or maybe not.)
Set 10
Evens Minus Odds
The sum of the first 80 positive odd integers subtracted from the sum of the first 80 positive even integers is:
- 0
- 20
- 40
- 60
- 80
Spying on Student Council
You have just been captured spying on a student council meeting, and you face dreadful consequences. You are told: “You may make one statement. If it is true, you will be dumped into a trash can in the cafeteria during the second lunch hour. If it is false, you will hang by your feet from the home basketball net during halftime of the varsity game this weekend.” What can your statement be so that you can save yourself from both of these terrifying fates?
Scales
The picture shows some equalities of weight among objects of four kinds: cylinders, spheres, cones, and cubes. At the bottom, four cones are placed in the left pan. What is the smallest number of objects we can put on the right pan to balance the cones?
Handshakes (24, 1)
Twenty-four people are at a party. Each person shakes hands once with each other person. How many handshakes are there?
Glenda Quiz
Glenda is taking a quiz with five questions on it. They are all true-false questions. She knows only these four things:
- there are more true answers than false answers;
- no 3 consecutive questions have the same answer;
- questions 1 and 5 have different answers;
- the answer to #2 (that is, she actually knows whether this one is true or false).
With only this much to go on, she got a perfect score. And so can you. What are the answers to the five questions?
Kant’s Clock, Neighbor
It is said that the philosopher Immanuel Kant was a man of exceptionally regular habits. He particularly liked to have his grandfather clock set at exactly the right time. One day he was distressed to find that his clock had run down. Evidently his manservant had forgotten to wind it. Kant restarted the clock but couldn’t reset the hands because his watch was at the repair shop and he had no way of knowing the correct time.
That evening he walked over to the house of his friend Meyer, a mile or so away, for their weekly chess game. Kant always walked at the same steady pace, though he had never taken note of how long this journey took him. He glanced at the grandfather clock as he was leaving and noted his arrival time on the clock in Meyer’s hallway. He played several games of chess, then returned home. He walked along the same route by which he had come, with the same regular gait that had not changed for years. He had no idea how long his trip took. When he arrived home, he immediately set his grandfather clock correctly.
From the information in this story, can you figure out the procedure Kant used so that he could set his clock correctly on returning home?
Old Watches
You work at the watch counter in Best’s. Two Swatch watches in the case are acting up, so you isolate them for study. You find that one of them loses two minutes per hour, and the other one gains one minute per hour. This means that in one hour of real time, one watch goes 58 minutes and the other goes 61 minutes. Now, you start both watches running at exactly twelve noon. If you look at them again when the faster watch is exactly one hour ahead of the slower watch, what time will it be in real time?
Crypto-Product
Here is a multiplication problem with some digits missing. Think a little and it is pretty easy to figure out what they are.
Cube, Orange, Cut Up
One thousand unit cubes are fastened together to form a large cube with edges of 10 units. The big cube is painted bright orange. After the paint is dry, the cube is taken apart into the original one thousand little cubes. How many of these unit cubes now have at least one orange face?